Needing Intuition in Math(s): one example

I’m helping a few people with abstract algebra at the moment, and I came to this realization. Most people learning abstract algebra, as far as I can tell, have no idea why homomorphisms and factor groups are sensible things to think about. They quickly come to understand the idea of a group, and enough varied examples are usually given that they can see how the idea of a group applies to a number of things. They quickly come to terms with subgroups, though the idea looks rather trivial to them. Then you get to homomorphisms and factor groups; at this point, most classes run out of intuition and just jump in for some unmotivated mathematical constructions.

I’m not quite sure why this is, honestly. Anyone with the slightest modicum of mathematical curiosity probably has thoughtn about factor groups since they were seven or eight years old. In the context of integers and addition, most children realize on their own (whether it’s taught to them or not) that the sum of two even numbers, or of two odd numbers, is even, while the sum of an even number and an odd number is odd. This is, of course, a factor group. Students who are presented with the mathematical definition of a factor group should first have, in their set of mental tools, this simple intuitive definition:

Factor Group: For any group (G,*), a factor group is a group that is obtained by being sufficiently sleep-deprived (or perhaps drunk, depending on the university) that one can’t tell the difference between some members of the original group, and then trying to write down a group table.

Of course, one then goes on to point out that sometimes this works, but sometimes it doesn’t. If one looks at the integers and only sees “even” or “odd”, then it works. If one looks at the integers and only sees “negative” or “non-negative”, then it doesn’t work, since the sum of a negative number and a positive number could be either negative or positive. It then becomes natural to ask when it works, and when it doesn’t. This provides a justification, then, for nailing down the abstract definitions, defining normal subgroups, proving that the factor group is well-defined when modding out a normal subgroup, and so on. First, though, the student needs to be convinced that these are natural things to think about.

(A quick aside: I’m not pulling out the idiotic canard that students need to be convinced that mathematics is useful in “the real world” or anything so ridiculous as that. But even the purest mathematicians are more interested in answering questions that naturally arise than those that seem quite arbitrary.)

Speaking of defining normal subgroups, it is really inexcusable how many students have never even noticed the close relationship between normal subgroups and commutativity. Sure, everyone knows that all subgroups of an abelian group are normal; but this seems to be treated as a sort of occasionally useful curiosity. Few students are even exposed to the simple fact that normality of subgroups is inextricably entwined in the degree to which the subgroup commutes with the surrounding group.

Case in point: one of the equivalent conditions for normality that is often taught is this: Let H be a subgroup of G. Then H is normal in G if and only if for any h in H, and any g in G, the product (g’ h g) is in H (where by g’, I mean the inverse of g). This statement is absolutely correct… but it is utterly useless to a student who will most certainly not recognize (g’ h g) as a statement about the commutativity of h and g. So first, it helps to show a few other results:

The following are equivalent: (g’ h g) = h, and gh = hg. The proof is trivial is both directions, and students can likely figure it out on their own. But, and this is the important part, students will likely not recognize this simple fact if they haven’t seen it before.

Let H be a subgroup of G. Then H is a subgroup of Center(G) if and only if for any h in H, and any g in G, (g’ h g) = h. This is just applying the previous result and the definition of a center.

From this point, the path to the earlier condition is clear. We’re really talking about commutativity, but of course our ultimate goal is to be able to blur our eyes (or get sleep-deprived, or drunk) such that we can’t tell the difference between the members of H. Then one may start with #2 above, and simply remove the distinction between members of H — so instead of looking for (g’ h g) = h, we look for (g’ h g) being “close enough” to h… in other words, it should be in the same subgroup. One may now recognize that normality really is a weaker analogue of commutativity.

So the literal point here is that students ought to be taught these two intuitions. The larger point is to wonder why it’s apparently considered appropriate to teach abstract algebra without teaching these two intuitions.

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3 Comments

I’m sure this is where you were going, but perhaps the ultimate way to think about normality as commutativity is that it expresses a condition of elements commuting with sets—in the sense that a subgroup H of G is normal if and only if, for all g in G, we have that gH = Hg. The analogous condition for commutativity of sets with one another gives the condition for the product of two groups to be a group—if H and K are subgroups of G, then HK := {hk : h in H, k in K} is a subgroup of G if and only if HK = KH.

As another example, I’ve often felt that one understands a good deal about abstract algebra, and indeed much of mathematics, if one gets the hang of when it’s OK to switch the order of words, and (because this usually indicates something profound) when it’s not. For example, I like to say to people that the condition for a map to be a homomorphism is that “the product of the images is the image of the product”—i.e., in Haskell, if T is the homomorphism and m is the (uncurried) multiplication map, T . m = m . (double T), where double T (x, y) = (Tx, Ty). (That double function probably already has a name, but I couldn’t find it.)

I’m not entirely sure I like the idea of normality being a weaker commutativity as you phrased it. You’re basically saying that H < G is normal if g^-1 h g is “close” to h in the sense of being in the same subgroup. Fine, but then my intuition would say that if H is a small subgroup then there are fewer options for g^-1 h g so it gets “closer” to h, and thus closer to being commutative, which I don’t think is a useful intuitive idea.

Perhaps a rephrasing along the following lines is useful: h and g commute if we can take the product hg and move g past h to give gh. H < G is normal if for any h in H, g in G, we can take the product hg and move g past the h to give gh’, where g’ is some other element of the subgroup.

FWIW, the way normal subgroups were justified in our course was along the lines of: we take two cosets gH and kH, and we’d like to find a way to multiply them, i.e. make the set of cosets G/H into a group. The natural way is of course to define (gh)(kH) = gkH — what conditions do we need for this to be well defined?